Hybrid Finite Element and Least Squares Support Vector Regression Method for solving Partial Differential Equations with Legendre Polynomial Kernels

TL;DR

The paper addresses the challenge of achieving high-accuracy PDE solutions without high-order FEM by fusing FEM with LSSVR using Legendre kernels.It introduces a three-step hybrid algorithm that uses FEM nodal values to constrain a localized LSSVR interpolation, yielding a closed-form, super-resolved solution.Across 1D and 2D Poisson and Helmholtz tests, the hybrid method consistently outperforms or matches high-order FEM with significantly reduced mesh complexity, and remains robust under larger domains and varied boundary conditions.The approach offers a plug-and-play tool for super-resolving expensive simulations and noisy measurements, with potential applicability to forward and inverse PDE problems.

Abstract

A hybrid computational approach that integrates the finite element method (FEM) with least squares support vector regression (LSSVR) is introduced to solve partial differential equations. The method combines FEM's ability to provide the nodal solutions and LSSVR with higher-order Legendre polynomial kernels to deliver a closed-form analytical solution for interpolation between the nodes. The hybrid approach implements element-wise enhancement (super-resolution) of a given numerical solution, resulting in high resolution accuracy, while maintaining consistency with FEM nodal values at element boundaries. It can adapt any low-order FEM code to obtain high-order resolution by leveraging localized kernel refinement and parallel computation without additional implementation overhead. Therefore, effective inference/post-processing of the obtained super-resolved solution is possible. Evaluation results show that the hybrid FEM-LSSVR approach can achieve significantly higher accuracy compared to the base FEM solution. Comparable accuracy is a achieved when comparing the hybrid solution with a standalone FEM result with the same polynomial basis function order. The convergence studies were conducted for four elliptic boundary value problems to demonstrate the method's ability, accuracy, and reliability. Finally, the algorithm can be directly used as a plug-and-play method for super-resolving low-order numerical solvers and for super-resolution of expensive/under-resolved experimental data.

Figures (4)

• Figure 1: Example 1 - 1D Poisson Equation Results: Comparison of fine mesh (many FEM nodes), coarse mesh (fewer FEM nodes), and convergence analysis for domains [-1,1] (top row) and [-5,5] (bottom row)
• Figure 2: Example 2 – 2D Poisson Equation Results: Hybrid FEM-LSSVR solution, error field, and convergence analysis for domains [-1,1]$\times$[-1,1] (top row) and [-2,2]$\times$[-2,2] (bottom row).
• Figure 3: Example 3 - 2D Helmholtz Equation with Dirichlet BC results showing hybrid FEM-LSSVR solution, error field, and convergence analysis for $n=\frac{1}{2}$ (top row) and $n=1$ (bottom row).
• Figure 4: Example 4 - 2D Helmholtz Equation with Neumann BC results showing hybrid FEM-LSSVR solution, error field, and convergence analysis for Mode $(1, 1)$ (top row, $\xi = \pi\sqrt{2} + 0.1i$) and Mode $(2, 2)$ (bottom row, $\xi = 2\pi\sqrt{2} + 0.1i$).